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EP2395698A1 - Occultation du résultat de générateurs de nombres aléatoires dans des protocoles de génération de clés - Google Patents

Occultation du résultat de générateurs de nombres aléatoires dans des protocoles de génération de clés Download PDF

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Publication number
EP2395698A1
EP2395698A1 EP10165765A EP10165765A EP2395698A1 EP 2395698 A1 EP2395698 A1 EP 2395698A1 EP 10165765 A EP10165765 A EP 10165765A EP 10165765 A EP10165765 A EP 10165765A EP 2395698 A1 EP2395698 A1 EP 2395698A1
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European Patent Office
Prior art keywords
public key
key
private key
trusted party
private
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EP10165765A
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German (de)
English (en)
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EP2395698B1 (fr
Inventor
Scott A Vanstone
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Certicom Corp
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Certicom Corp
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Priority to EP20100165765 priority Critical patent/EP2395698B1/fr
Priority to CA2742530A priority patent/CA2742530C/fr
Publication of EP2395698A1 publication Critical patent/EP2395698A1/fr
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Publication of EP2395698B1 publication Critical patent/EP2395698B1/fr
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
    • H04L9/32—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols including means for verifying the identity or authority of a user of the system or for message authentication, e.g. authorization, entity authentication, data integrity or data verification, non-repudiation, key authentication or verification of credentials
    • H04L9/3263—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols including means for verifying the identity or authority of a user of the system or for message authentication, e.g. authorization, entity authentication, data integrity or data verification, non-repudiation, key authentication or verification of credentials involving certificates, e.g. public key certificate [PKC] or attribute certificate [AC]; Public key infrastructure [PKI] arrangements
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
    • H04L9/30—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy
    • H04L9/3006—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy underlying computational problems or public-key parameters
    • H04L9/302—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy underlying computational problems or public-key parameters involving the integer factorization problem, e.g. RSA or quadratic sieve [QS] schemes
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
    • H04L9/30—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy
    • H04L9/3066—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy involving algebraic varieties, e.g. elliptic or hyper-elliptic curves
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L2209/00—Additional information or applications relating to cryptographic mechanisms or cryptographic arrangements for secret or secure communication H04L9/00
    • H04L2209/04—Masking or blinding

Definitions

  • the following relates generally to the field of cryptography.
  • Public key cryptographic protocols were first proposed in 1976 by Diffie-Hellman. A public-private key pair is created for each correspondent, with the public key made available to other parties, and the private key information maintained secret by the correspondent who will be the recipient of messages. Any message encrypted using the public key of a recipient can only be decrypted using the private key of that same recipient. The private key cannot be derived from the knowledge of the plaintext, ciphertext and public-key.
  • Cryptographic systems utilise protocols that are based on so called “hard” problems. These problems can be formulated quickly, but do not have efficient solution algorithms. Problems such as integer factorization and the discrete logarithm problem fall into this category.
  • Integer factorisation is the basis of a set of protocols known as RSA which uses, as a modulus n, the product of two large primes, p, q.
  • a second modulus (D is computed as (p-1)(q-1).
  • a correspondent's public key is then ( n,e ) and the corresponding private key is d.
  • the recipient's public key (n,e) is obtained and the message represented as an integer m in the interval [0, n-1 ].
  • the RSA algorithm is a deterministic algorithm requiring the selection of the large primes p, q.
  • integer factorisation techniques have improved, it has become necessary to use larger moduli, and, consequently, the computational efficiency has decreased.
  • the discrete log problem forms the basis for discrete log cryptographic systems that include Diffie Hellman key agreement protocols and ElGamal encryption and signature schemes.
  • a corresponding public key is computed as ⁇ x and distributed publically to other correspondents.
  • the public and private keys are used according to well known protocols to encrypt and decrypt messages sent between parties, to authenticate a message signed by one party using a private key, or to establish a common key between the parties by combining the public key of one party with the private key of another.
  • elliptic curve cyclic group defined over the finite field F p composed of integers from 0 to p-1 , where p is a prime number.
  • the group elements are points lying on a defined elliptic curve and having coordinates that are elements of the underlying field.
  • An elliptic curve group typically utilises an additive notation, rather than the multiplicative notation used above, so that a k -fold group operation of a point P requires the point P to be added k times and is written kP.
  • a cryptographic system implemented using an elliptic curve group is known as an elliptic curve cryptographic system, or ECC.
  • ECC elliptic curve cryptographic system
  • Other groups commonly used are multiplicative groups, such as the non-zero integers F p and the corresponding k -fold group operation is denoted ⁇ k , where ⁇ is a generator of the group.
  • a random number generator (RNG) is designed to provide cryptographically strong random numbers, but, due to malfunction, poor selection of a seed, or malicious tampering, the RNG may output relatively weak random numbers. This is a particular problem in constrained devices, such as cell phones and smart cards, where low cost implementations may not have sufficient entropy to provide a robust random number generator. As a result, publication of the public key may inadvertently disclose the corresponding private key, and, depending on the protocol implemented, may in turn yield information pertaining to other private keys used in the protocol.
  • the following may provide a technique for masking the output of a random number generator (RNG).
  • RNG random number generator
  • the output of this RNG may be used by a correspondent as a private value, for example, an ephemeral private key.
  • the correspondent may then generate a corresponding public key.
  • direct scrutiny of this public key (which could lead to exposure of the corresponding private value due to a weakness of the RNG) may be avoided during transmission by encrypting the public key with a deterministic encryption scheme, such as RSA, before distributing the public key in a public manner.
  • the deterministic encryption scheme does not rely on the output of the RNG for its strength and therefore may effectively mask the public key and the private value used to generate the public key.
  • the encrypted public key may be received by a trusted party who can decrypt it and recover the public key.
  • the trusted party may utilise the public key in combination with other information to avoid direct exposure of the public key to third parties.
  • the trusted party Preferably, the trusted party generates an implicit certificate that incorporates, but does not directly expose, the public key. The implicit certificate can then be used by the correspondent in communications with other correspondents.
  • a computer readable medium There is also preferably provided a computing device for use in a cryptographic system according to claim 8.
  • a computer readable medium comprising computer executable instructions which when executed cause the computer to perform the method of any of claim 1 and its dependent claims.
  • a method of one computing device acting as a trusted party, generating a certificate, for use by another correspondent in a public key cryptographic system according to claim 12.
  • a computer readable medium There is also preferably provided a computing device for use in a cryptographic system configured to perform the steps of the method of claim 12.
  • a computer readable medium comprising computer executable instructions which when executed cause the computer to perform the method of any of claim 12 and its dependent claims.
  • Figure 1 is a schematic representation of a data communication system.
  • Figure 2 is a representation of a cryptographic unit used in the system of Figure 1 .
  • Figure 3 is a schematic representation of the steps performed by the system of Figure 1 .
  • Figure 4 is a schematic representation of the flow of information within the system of Figure 1 .
  • Figure 5 is a schematic representation of a method of establishing a private key from the perspective of the requestor.
  • Figure 6 is a schematic representation of the method of Figure 5 from the perspective of the trusted party.
  • a data communication system includes a pair of correspondents 12, 13 connected by a communication link 16.
  • the correspondents 12, 13, are connected through link 16 to a mutually trusted correspondent 14, that is referred to as a certifying authority (CA).
  • the correspondents 12, 13 are computing devices, such as a personal computer, personal digital assistant, smart phone, cellular phone, ATM, Point of Sale device, server, entertainment system component, or other such device having a computing capability and the ability to exchange digital information with other correspondents.
  • the CA 14 is also a computing device, typically a server with a secure module for maintaining keys and performing cryptographic processes.
  • the CA 14 is itself connected to higher level CA's providing a trusted hierarchy in well known manner.
  • the communication link 16 may be a telephone link, wireless or landline, local area network (LAN), RF link, or other such link provided to transfer information between the correspondents. Although shown for illustrative purposes as direct connection between correspondents 12, 13 and CA 14, it will be appreciated that the communication link 16 may be part of a more extensive network and that the link may be routed through multiple correspondents in passing from correspondent 12 to correspondent 13 or CA 14.
  • LAN local area network
  • correspondent 12 includes a cryptographic unit 18 that communicates with a memory 20 and a processor 22.
  • the correspondent may also include a data input unit 24, such as a keypad or card reader, and a display device, 26, depending on the intended purpose of the correspondent 12.
  • the cryptographic unit 18 is provided to manage secure communications between the correspondents 12, 13 and CA 14 over the communication link 16.
  • the cryptographic unit 18 includes a secure memory 30, which may be part of the memory 20 or a separate memory module, and an arithmetic logic unit (ALU), 32, that operates under the control of the processor 22 to perform arithmetic operations necessary to implement a chosen cryptographic protocol.
  • the processor 22 includes one or more instruction sets to implement respective protocols.
  • any module or component exemplified herein that executes instructions may include or otherwise have access to non-transient computer readable media such as storage media, computer storage media, or data storage devices (removable and/or non-removable) such as, for example, magnetic disks, optical disks, or tape.
  • Computer storage media may include volatile and non-volatile, removable and non-removable media implemented in any method or technology for storage of information, such as computer readable instructions, data structures, program modules, or other data.
  • Examples of computer storage media include RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can be accessed by an application, module, or both. Any such computer storage media may be part of the cryptographic unit or accessible or connectable thereto. Any application or module herein described may be implemented using computer readable/executable instructions that may be stored or otherwise held by such computer readable media.
  • the memory 30 stores the parameters of the cryptographic schemes and protocols to be implemented by the cryptographic unit 18.
  • the cryptographic unit 18 may execute a program to implement a public key elliptic curve cryptographic system in which cryptographic protocols are implemented based on the intractability of the discrete log problem in an elliptic curve group defined over a finite field.
  • the memory 30 therefore stores the generator point G, the parameters of the curve, and the order, n , of the elliptic curve group G .
  • the cryptographic unit 18 may also execute a deterministic cryptographic scheme, typically a RSA integer factorisation scheme, although other deterministic schemes could be used.
  • the memory 30 therefore stores a private key, d , that has a corresponding public key (n, e ).
  • the memory 30 also stores the RSA public key (n', e') of the correspondent 13 and the public key ( n", e ") of the CA 14.
  • the cryptographic unit 18 may also execute a program to implement a symmetric encryption protocol utilising a shared secret key.
  • the cryptographic unit 18 also includes a random number generator 34, whose output is provided in a secure manner to memory 30 for use by the ALU 32.
  • the ALU 32 uses the bit string as an input to a key derivation function (KDF) that outputs a session private key x , that is a bit string derived from the output of the random number generator 34.
  • KDF key derivation function
  • the session key x is stored securely in memory 30 and used by the ALU 32 in generating an ephemeral or session public key xG .
  • the computation is performed in the ALU 32 using point multiplication algorithms and the bit string representing the private key x . Suitable algorithms are well known in the art and described at section 3.3 of the Guide to Elliptic Curve Cryptography, Hankerson et al., Springer-Verlag, 2004 and so need not be described in further detail.
  • each of the correspondent 13 and CA 14 securely store RSA private keys (d'), (d") respectively and distribute their public keys (n', e') (n", e").
  • the discrete log based protocols typically require the generation of key pair for each session and hence the use of the RNG 34 for each session.
  • the deterministic encryption scheme is used to blind the public key until such time as the ephemeral public key is masked by other computations.
  • the correspondents 12, 13 implement via the link 16, an implicit certificate protocol using the respective cryptographic unit 18.
  • An implicit certificate protocol enables two correspondents to communicate without the recipient explicitly verifying a signature on a certificate of the sender's public key from the certification authority CA.
  • the recipient receives an implicit certificate of the sender from which the recipient may recover the sender's ephemeral public key.
  • the initial step of such a protocol is to perform a certificate enrolment between the sender and the CA.
  • the sequence of steps and the flow of information is shown in Figures 3 and 4 respectively.
  • Each of the correspondents 12, 13 has, in its memory 30, a copy of the long term or static public key Q CA of the CA 14, and the CA 14 has the corresponding private key k CA securely stored in its memory 30.
  • each of the correspondents 12, 13 has the RSA public key of the CA 14 stored in memory.
  • the correspondent 12 initiates the certificate enrolment by generating a random bit string at the output of the RNG 34 as indicated at step 102.
  • the cryptographic unit uses the RNG output as an input to a key derivation function, which may simply be a truncation of a bit string, to obtain a private value that in the illustrated embodiment is an ephemeral private key x A .
  • the cryptographic unit 18 uses, at step 108, the RSA public key (n",e") of the CA 14 to encrypt the ephemeral public key X A and obtain the encrypted message X A '.
  • the encrypted message X A ', together with the identifying information, ID A , of the correspondent 12 is sent over the link 16 to the CA 14 (step 110).
  • the CA 14 uses its RSA private, d", key to decrypt the message X A ' and recover the ephemeral public key X A . It will be apparent that any third party intercepting the message from the correspondent 12 will be unable to recover public key X A as the RSA private key remains secret to the CA 14. Moreover, the use of the deterministic cryptographic scheme does not require the use of the RNG 34, and therefore avoids the potential vulnerability if a weak RNG 34 is used.
  • the cryptographic unit 18 of the CA 14 performs a validation to ensure the key X A satisfies the necessary criteria for a public key, e.g. that it is a point on the elliptic curve and is not an element of a small subgroup or the point at infinity.
  • the identity information, ID A is validated for content (step 114) and if both are found satisfactory the cryptographic unit 18 of the CA 14 proceeds to the next step, indicated at 116, and uses the RNG 34 in the CA 14 to generate an ephemeral private key x CA .
  • the ephemeral public keys X A and X CA are combined, as at step 118, to obtain a new point B A which is the public key reconstructions data for correspondent 12.
  • the point B A is concatenated with the identification ID A of the correspondent 12, to provide an implicit certificate IC A (step 120).
  • the private key contribution data, s is generated by the cryptographic unit 18. This is done by hashing the certificate IC A using a cryptographically secure hash, such as SHA2, to obtain a value e .
  • the cryptographic unit 18 combines the values from the memory 30 performing integer arithmetic and modular reduction.
  • the shared key R is used by the cryptographic unit to encrypt the private key reconstruction data s using a symmetric encryption scheme to obtain the encrypted message s'.
  • the encrypted message s ' and the implicit certificate IC A is then sent to the correspondent 12 over the link 16 (step 126)
  • the correspondent 12 initially parses the certificate IC A to obtain the public key reconstruction data, namely point B A , and the identity information ID A (step 128).
  • the identity information ID A and point B A are validated (step 130) and the shared key R obtained by computing x A Q CA , as indicated at 132.
  • the shared key R is used to decrypt the message s' and obtain s (step 134) from which the private key k A associated with the implicit certificate IC A can be obtained.
  • the private key k A is obtained from ex A + s (mod n ), where e is the hash of the certificate IC A (step 136). It will be noted that generation of R and k A requires the use of the ephemeral private key x A which is only known to the correspondent 12.
  • the public key Q A can therefore be computed by any correspondent, e.g. correspondent 13, who receives the implicit certificate IC A and has the public key of the CA 14.
  • the correspondent 12 may use the key pair k A , Q A and for example sign the information using k A .
  • the recipient 13 receives the signed information and the implicit certificate IC A and can recover the public key Q A by computing e from the certificate IC A , parsing B A from the certificate IC A , and using the public key Q CA of the CA 14.
  • the public key X A obtained from the private key x A is not directly exposed to scrutiny.
  • the public key X A is blinded by the encryption using a deterministic scheme which does not utilise the RNG 34.
  • the implicit certificate IC A provides the point B A , which is a combination of the ephemeral keys X A and X CA .
  • the ephemeral key of the CA 14 is generated by the RNG of the CA, which can be assumed to provide an output that is cryptographically acceptable. Therefore the combination of the potentially vulnerable public key X A with the secure public key X CA masks the public key X A and inhibits retrieval of the private key x A .
  • the shared key R which uses the private key x A , is maintained secret and therefore does not expose the private key x A to allow others to reconstruct the shared key R .
  • FIG. 3 and 4 The embodiment shown in Figures 3 and 4 is only one specific embodiment. More generally, a method is contemplated for establishing a private key for use in a public key cryptographic system. The method is described with reference Figures 5 and 6 , and it will be appreciated that the method can be performed by computing devices configured to perform the operations, or that Figures 5 and 6 can comprise a set of computer readable instructions executed by the computing devices.
  • step 202 the device wishing to establish the private key obtains a private value from an output of the random number generator 34. Then, in step 204, a corresponding public key is computed from the private value.
  • step 206 the corresponding public key is encrypted using a deterministic encryption scheme to establish an encrypted public key, and in step 208 the encrypted public key is forwarded to a computing device acting as a trusted party.
  • the device receives from the trusted party encrypted private key reconstruction data.
  • a decryption key is used to recover the private key reconstruction data, and in step 214, the private key is computed from the private key reconstruction data.
  • the private key is a discrete log private key corresponding to a discrete log public key made publicly available. This private key may be subsequently used in a transfer of information with a recipient of the corresponding discrete log public key.
  • Figure 6 describes the method of Figure 5 from the perspective of the trusted party.
  • the trusted party receives from the device the encrypted public key transmitted in step 208.
  • the trusted party decrypts the public key using the deterministic encryption scheme to obtain the public key.
  • step 306 the trusted party utilizes the public key to generate the private key reconstruction data, and it is encrypted in step 308.
  • step 310 the trusted party forwards the encrypted private key reconstruction data to the device, which uses it as described in steps 210 to 214.

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EP20100165765 2010-06-11 2010-06-11 Génération de certificats implicites dans le contexte de générateurs aléatoires faibles Active EP2395698B1 (fr)

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EP20100165765 EP2395698B1 (fr) 2010-06-11 2010-06-11 Génération de certificats implicites dans le contexte de générateurs aléatoires faibles
CA2742530A CA2742530C (fr) 2010-06-11 2011-06-10 Masquage de la sortie des generateurs de nombres aleatoires dans les protocoles de generation de cles cryptographiques

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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
WO2014205570A1 (fr) * 2013-06-27 2014-12-31 Infosec Global Inc. Protocole d'agrément de clé
WO2016187690A1 (fr) * 2015-05-26 2016-12-01 Infosec Global Inc. Protocole d'agrément de clé
CN113965324A (zh) * 2021-12-07 2022-01-21 国家信息技术安全研究中心 基于模板的rsa-crt实现模约减攻击的私钥恢复方法和系统

Families Citing this family (1)

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Publication number Priority date Publication date Assignee Title
FR3095709B1 (fr) * 2019-05-03 2021-09-17 Commissariat Energie Atomique Procédé et système de masquage pour la cryptographie

Citations (1)

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US20050114651A1 (en) * 1998-03-23 2005-05-26 Minghua Qu Implicit certificate scheme

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US20050114651A1 (en) * 1998-03-23 2005-05-26 Minghua Qu Implicit certificate scheme

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Title
BROWN D R L ET AL: "Provably secure implicit certificate schemes", FINANCIAL CRYPTOGRAPHY, vol. 2339, 1 January 2002 (2002-01-01), pages 156 - 165, XP002524293, ISBN: 978-3-540-24128-7 *
EUN-JUN YOON ET AL: "An Optimizing Authenticated Key Exchange Protocol for Self-organizing Sensor Networks", 1 January 2006, UBIQUITOUS COMPUTING SYSTEMS LECTURE NOTES IN COMPUTER SCIENCE;;LNCS, SPRINGER, BERLIN, DE, PAGE(S) 537 - 546, ISBN: 978-3-540-46287-3, XP019045631 *
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Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
WO2014205570A1 (fr) * 2013-06-27 2014-12-31 Infosec Global Inc. Protocole d'agrément de clé
US9571274B2 (en) 2013-06-27 2017-02-14 Infosec Global Inc. Key agreement protocol
WO2016187690A1 (fr) * 2015-05-26 2016-12-01 Infosec Global Inc. Protocole d'agrément de clé
CN113965324A (zh) * 2021-12-07 2022-01-21 国家信息技术安全研究中心 基于模板的rsa-crt实现模约减攻击的私钥恢复方法和系统

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EP2395698B1 (fr) 2014-08-13
CA2742530C (fr) 2016-12-20

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